Monomials, Polynomials

Date: 03/02/2001 at 17:57:10
From: Doctor Jordi
Subject: Re: Monomials
Hello, Chris - thanks for writing to Dr. Math.
I am not sure exactly what you would like to know about monomials and
polynomials. There is not much to say about monomials, except that
they are the building blocks of polynomials. Polynomials, on the other
hand, are very interesting mathematical structures, and a whole lot of
things can be said about them. I will spare you for the moment saying
every interesting fact about polynomials that I can think of, since I
am imagining that for now you are only interested in knowing the
definitions regarding the two.
A monomial is a product of as many factors as you like, each raised to
a POSITIVE power. By definition, negative exponents are not allowed.
The following are examples of monomials:
x
y
x^2 (x squared)
3xy
34.6*(a^3)*r^4
3(x^26)*(y^(pi)) (3 times x to the 26th power times y to the pi power)
37*a*b*c*d* ... *z*alpha*beta*gamma* ... *omega*aleph*(a partridge in
a pear tree)
(The product of 37 times the variables represented by all the letters
of the English alphabet times the variables represented by all the
letters of the Greek alphabet times the first letter of the Hebrew
alphabet, times a partridge in a pear tree. It is still a monomial, no
matter how many numbers we are multiplying together. Don't worry too
much about this goofy example; I am just trying to point out that a
monomial is a very general concept.)
A polynomial is nothing more than the sum of two or more monomials.
The following are examples of polynomials:
x (yes, a monomial is also a polynomial)
x + y
ax^2 + bx + c (the very famous quadratic polynomial)
(x^2)*(y^2) + xy + x
x^34 + r^(3.535) + c
abc + rst + xyz^2 + abcdefghijklmnopqrstuvwxyz
Again, don't worry too much about that last goofy example. I'm sure
you will probably never encounter a polynomial anytime soon that uses
all the letters of the English alphabet.
Usually, however, when we say "polynomial," we are talking about a
very specific kind of polynomial that occurs very often. This is the
polynomial where there is only one variable, all the powers of this
variable are nonnegative integers, and each power of this variable may
be multiplied by a constant. The way mathematicians usually write this
polynomial is
A_n * x^n + A_(n-1) * x^(n-1) + ... + A_2 * x^2 + A_1 * x + A_0
Where the A_i symbols (A_i is intended to represent A with a subscript
i, the letter A indexed by i) represent constants (i.e. numbers like
0, 1, -3, 7/4, 34.334 and pi) and n is a positive integer. The famous
quadratic polynomial I mentioned above in my examples is a special
case of these very common polynomials when n = 2.
In these polynomials, by the way, n is called the "degree" of the
polynomial. Thus, a quadratic polynomial has degree 2. Polynomials of
degree 3 are called "cubic polynomials," of degree 1 are called
"linear expressions" (they are technically also polynomials, but
nobody calls them that) and of degree 0 (i.e., there are no variables,
only the A_0) are called "constants." Yes, constants are also a
special case of polynomials. In fact, a constant by itself is also a
monomial, but we usually never call them that, in order to avoid
confusion.
Summarizing, a monomial is a single mathematical expression that
contains only multiplication and exponentiation. A polynomial is a sum
of monomials, so you can think of it as a mathematical expression that
only contains the operations of addition, multiplication, and
exponentiation.
I hope this explanation helped. Please write back if you have more
questions, or if you would like to talk about this some more.
- Doctor Jordi, The Math Forum
http://mathforum.org/dr.math/